Complex asinh accuracy refinement

libcudacxx/include/cuda/std/__complex/inverse_hyperbolic_functions.h

@@ -39,51 +39,360 @@
 
 _CCCL_BEGIN_NAMESPACE_CUDA_STD
 
+// Specialization of 1/sqrt(x), uses device-only rsqrtf/rsqrt when viable.
+// Sometimes compilers can optimize 1/sqrt(x) so we don't suffer the slowdown of the divide.
+// It can happen that the double 1/sqrt() on device gets optimized better than CUDA rsqrt(),
+// but it depends on the calling function. We needs to optimize on an individual basis.
+template <class _Tp>
+[[nodiscard]] _CCCL_API inline _Tp __internal_rsqrt_inverse_hyperbloic(_Tp __x) noexcept
+{
+#if _CCCL_CUDA_COMPILATION()
+  if constexpr (is_same_v<_Tp, float>)
+  {
+    NV_IF_TARGET(NV_IS_DEVICE, (return ::rsqrtf(__x);))
+  }
+  if constexpr (is_same_v<_Tp, double>)
+  {
+    NV_IF_TARGET(NV_IS_DEVICE, (return ::rsqrt(__x);))
+  }
+#endif // _CCCL_CUDA_COMPILATION()
+  return _Tp{1} / ::cuda::std::sqrt(__x);
+}
+
+template <class _Tp>
+struct _CCCL_ALIGNAS(2 * sizeof(_Tp)) __cccl_asinh_sqrt_return_hilo
+{
+  _Tp __hi;
+  _Tp __lo;
+};
+
+// An unsafe sqrt(_Tp + _Tp) extended precision sqrt.
+template <typename _Tp>
+static _CCCL_API inline __cccl_asinh_sqrt_return_hilo<_Tp>
+__internal_double_Tp_sqrt_unsafe(_Tp __hi, _Tp __lo) noexcept
+{
+  // rsqrt
+  const _Tp __initial_guess = __internal_rsqrt_inverse_hyperbloic<_Tp>(__hi);
+
+  // Newton-Raphson, assume we have a reasonable rsqrt above and
+  // that we don't need to update the first term (__initial_guess).
+  //     x_(n+1) = x_n - 0.5*x_n*((__hi + __lo)(x_n^2) - 1)
+
+  // __initial_guess^2:
+  const _Tp __init_sq_hi = __initial_guess * __initial_guess;
+  const _Tp __init_sq_lo = ::cuda::std::fma(__initial_guess, __initial_guess, -__init_sq_hi);
+
+  // Times (__hi + __lo).
+  // We need to add the -1.0 here in an fma, or different compilers (eg host vs device)
+  // optimize differently and give (sometimes very) different results.
+  const _Tp _hi_hi_hi = ::cuda::std::fma(__hi, __init_sq_hi, _Tp{-1.0});
+  // Low part not needed for fp32/fp64, but might be if this is extended to other types:
+  // _Tp _hi_hi_lo = fma(__hi, __init_sq_hi, -_hi_hi_hi);
+
+  // Add all terms
+  const _Tp __full_term = _hi_hi_hi + (::cuda::std::fma(__lo, __init_sq_hi, __hi * __init_sq_lo) /*+ _hi_hi_lo*/);
+
+  const _Tp __correction_term = _Tp(-0.5) * __initial_guess * __full_term;
+
+  // rsqrt(hi + lo) is now estimated well by (__initial_guess + __correction_term)
+  // Multiply everything by (hi + lo) to get sqrt(hi + lo)
+  const _Tp __ans_hi_hi = __hi * __initial_guess;
+  _Tp __ans_hi_lo       = ::cuda::std::fma(__hi, __initial_guess, -__ans_hi_hi);
+
+  // All terms needed, allow the compiler to pick which way to
+  // optimize this to fma, same accuracy.
+  __ans_hi_lo += __initial_guess * __lo + __correction_term * __hi;
+
+  __cccl_asinh_sqrt_return_hilo<_Tp> __retval_hilo;
+  __retval_hilo.__hi = __ans_hi_hi;
+  __retval_hilo.__lo = __ans_hi_lo;
+
+  return __retval_hilo;
+}
+
 // asinh
 
 template <class _Tp>
 [[nodiscard]] _CCCL_API inline complex<_Tp> asinh(const complex<_Tp>& __x)
 {
-  constexpr _Tp __pi = __numbers<_Tp>::__pi();
-  if (::cuda::std::isinf(__x.real()))
+  // Uint of the same size as our fp type.
+  using __uint_t = __fp_storage_of_t<_Tp>;
+
+  constexpr int32_t __mant_nbits = __fp_mant_nbits_v<__fp_format_of_v<_Tp>>;
+  constexpr int32_t __exp_max    = __fp_exp_max_v<__fp_format_of_v<_Tp>>;
+
+  constexpr _Tp __pi  = __numbers<_Tp>::__pi();
+  constexpr _Tp __ln2 = __numbers<_Tp>::__ln2();
+
+  _Tp __realx = ::cuda::std::fabs(__x.real());
+  _Tp __imagx = ::cuda::std::fabs(__x.imag());
+
+  // Special cases that do not pass through:
+  if (!::cuda::std::isfinite(__realx) || !::cuda::std::isfinite(__imagx))
   {
-    if (::cuda::std::isnan(__x.imag()))
+    // If z is (x,+inf) (for any positive finite x), the result is (+inf, pi/2)
+    if (::cuda::std::isfinite(__realx) && ::cuda::std::isinf(__imagx))
     {
-      return __x;
+      return complex<_Tp>(::cuda::std::copysign(numeric_limits<_Tp>::infinity(), __x.real()),
+                          ::cuda::std::copysign(_Tp(0.5) * __pi, __x.imag()));
     }
-    if (::cuda::std::isinf(__x.imag()))
+
+    // If z is (+inf,y) (for any positive finite y), the result is (+inf,+0)
+    if (::cuda::std::isinf(__realx) && ::cuda::std::isfinite(__imagx))
     {
-      return complex<_Tp>(__x.real(), ::cuda::std::copysign(__pi * _Tp(0.25), __x.imag()));
+      return complex<_Tp>(::cuda::std::copysign(numeric_limits<_Tp>::infinity(), __x.real()),
+                          ::cuda::std::copysign(_Tp(0), __x.imag()));
     }
-    return complex<_Tp>(__x.real(), ::cuda::std::copysign(_Tp(0), __x.imag()));
-  }
-  if (::cuda::std::isnan(__x.real()))
-  {
-    if (::cuda::std::isinf(__x.imag()))
+
+    // If z is (+inf,+inf), the result is (+inf, pi/4)
+    if (::cuda::std::isinf(__realx) && ::cuda::std::isinf(__imagx))
     {
-      return complex<_Tp>(__x.imag(), __x.real());
+      return complex<_Tp>(::cuda::std::copysign(numeric_limits<_Tp>::infinity(), __x.real()),
+                          ::cuda::std::copysign(_Tp(0.25) * __pi, __x.imag()));
     }
-    if (__x.imag() == _Tp(0))
+
+    // If z is (+inf,NaN), the result is (+inf,NaN)
+    if (::cuda::std::isinf(__realx) && ::cuda::std::isnan(__imagx))
     {
       return __x;
     }
-    return complex<_Tp>(__x.real(), __x.real());
+
+    // If z is (NaN,+0), the result is (NaN,+0)
+    if (::cuda::std::isnan(__realx) && (__imagx == _Tp{0}))
+    {
+      return __x;
+    }
+
+    // If z is (NaN,+inf), the result is (±INF,NaN) (the sign of the real part is unspecified)
+    if (::cuda::std::isnan(__realx) && ::cuda::std::isinf(__imagx))
+    {
+      return complex<_Tp>(__x.imag(), numeric_limits<_Tp>::quiet_NaN());
+    }
   }
-  if (::cuda::std::isinf(__x.imag()))
+
+  // Special case that for various reasons does not pass
+  // easily through the algorithm below:
+  if ((__realx == _Tp{0}) && (__imagx == _Tp(1)))
   {
-    return complex<_Tp>(::cuda::std::copysign(__x.imag(), __x.real()),
-                        ::cuda::std::copysign(__pi / _Tp(2), __x.imag()));
+    return complex<_Tp>(__x.real(), ::cuda::std::copysign(_Tp{0.5} * __pi, __x.imag()));
   }
-  complex<_Tp> __z = ::cuda::std::log(__x + ::cuda::std::sqrt(::cuda::std::__sqr(__x) + _Tp(1)));
-  return complex<_Tp>(::cuda::std::copysign(__z.real(), __x.real()), ::cuda::std::copysign(__z.imag(), __x.imag()));
+
+  // It is a little involved to account for large inputs in an inlined-into-existing-code fashion.
+  // The easiest place to account for large values appears to be here at the start.
+  // Use asinh(x) ~ log(2x) for large x.
+  // Get the largest exponent that passes through the algorithm without issue.
+  // This is ~(max_exponent / 4), with a small bias to make sure edge cases get caught
+  // ~254 for double, ~30 for float
+  constexpr int32_t __max_allowed_exponent     = (__exp_max / 4) - 2;
+  constexpr __uint_t __max_allowed_val_as_uint = __uint_t(__max_allowed_exponent + __exp_max) << __mant_nbits;
+
+  //  Check if the largest component of __x is > 2^__max_allowed_exponent:
+  _Tp __x_big_factor = _Tp{0};
+  const _Tp __max    = (__realx > __imagx) ? __realx : __imagx;
+  const bool __x_big = ::cuda::std::bit_cast<__uint_t>(__max) > __max_allowed_val_as_uint;
+
+  if (__x_big)
+  {
+    // We need __max to be <= ~(2^__max_allowed_exponent),
+    // but not small enough that the asinh(x) ~ log(2x) estimate does
+    // not break down. We are not able to reduce this with a single simple reduction,
+    // so we do a fast/inlined frexp/ldexp:
+    const int32_t __exp_biased = int32_t(::cuda::std::bit_cast<__uint_t>(__max) >> __mant_nbits);
+
+    // Get a factor such that (__max * __exp_mul_factor) <= __max_allowed_exponent
+    const __uint_t __exp_reduce_factor =
+      __uint_t((2 * __exp_max) + __max_allowed_exponent - __exp_biased) << __mant_nbits;
+    const _Tp __exp_mul_factor = ::cuda::std::bit_cast<_Tp>(__exp_reduce_factor);
+
+    // Scale down to a working range.
+    __realx *= __exp_mul_factor;
+    __imagx *= __exp_mul_factor;
+
+    __x_big_factor = _Tp((__exp_biased - __exp_max) - __max_allowed_exponent) * __ln2;
+  }
+
+  // let compiler pick which way to fma this, accuracy stays the same.
+  const _Tp __diffx_m1 = __realx * __realx - (__imagx - _Tp{1}) * (__imagx + _Tp{1});
+
+  // Get the real and imag parts of |sqrt(z^2 + 1)|^2
+  // This equates to calculating:
+  //     sqrt((re*re + (im + 1.0)*(im + 1.0))*(re*re + (im - 1.0)*(im - 1.0)));
+  // Where we need both the term inside the sqrt in extended precision, as well
+  // as evaluation the sqrt itself in extended precision.
+
+  // Get re^2 + im^2 + 1 in extended precision.
+  // The low part of re^2 doesn't seem to matter.
+  const _Tp __imagx_sq_hi = __imagx * __imagx;
+  const _Tp __imagx_sq_lo = ::cuda::std::fma(__imagx, __imagx, -__imagx_sq_hi);
+
+  const _Tp __x_abs_sq_hi = __imagx_sq_hi;
+  const _Tp __x_abs_sq_lo = ::cuda::std::fma(__realx, __realx, __imagx_sq_lo);
+
+  // Add one:
+  const _Tp __x_abs_sq_p1_hi = (__x_abs_sq_hi + _Tp{1});
+  const _Tp __x_abs_sq_p1_lo = __x_abs_sq_lo - ((__x_abs_sq_p1_hi - _Tp{1}) - __x_abs_sq_hi);
+
+  // square:
+  const _Tp __x_abs_sq_p1_sq_hi = __x_abs_sq_p1_hi * __x_abs_sq_p1_hi;
+  _Tp __x_abs_sq_p1_sq_lo       = ::cuda::std::fma(__x_abs_sq_p1_hi, __x_abs_sq_p1_hi, -__x_abs_sq_p1_sq_hi);
+
+  // Add in the lower square terms, all needed
+  __x_abs_sq_p1_sq_lo =
+    ::cuda::std::fma(__x_abs_sq_p1_lo, (_Tp{2} * __x_abs_sq_p1_hi + __x_abs_sq_p1_lo), __x_abs_sq_p1_sq_lo);
+
+  // Get __x_abs_sq_p1_sq_hi/lo - 4.0*__imagx_sq_hi/lo:
+  // Subtract high parts:
+  _Tp __inner_most_term_hi = __x_abs_sq_p1_sq_hi - _Tp{4} * __imagx_sq_hi;
+  _Tp __inner_most_term_lo = ((__x_abs_sq_p1_sq_hi - __inner_most_term_hi) - _Tp{4} * __imagx_sq_hi);
+  // lo parts, all needed:
+  __inner_most_term_lo += __x_abs_sq_p1_sq_lo - _Tp{4} * __imagx_sq_lo;
+
+  // We can have some slightly bad cases here due to catastrohip cancellation that can't be fixed easily.
+  // We still need to to the extended-sqrt on these values, so we fix them now.
+  // It occurs around "real ~= small" and "imag ~= (1 - small)", and imag < 1.
+  // Worked out through targeted testing on fp64 and fp32.
+  _Tp __realx_small_bound = _Tp{1.0e-13};
+  _Tp __imagx_close_bound = _Tp{0.98};
+
+  if constexpr (is_same_v<_Tp, float>)
+  {
+    __realx_small_bound = _Tp{1.0e-5f};
+    __imagx_close_bound = _Tp{0.9f};
+  }
+
+  if ((__realx < __realx_small_bound) && (__imagx_close_bound < __imagx) && (__imagx <= _Tp{1}))
+  {
+    // Get (real^2 + (1 - imag)^2) * (real^2 + (1 + imag)^2) in double-double:
+    // term1 = (real^2 + (1 - imag)^2)
+    const _Tp __term1_hi = (_Tp{1} - __imagx) * (_Tp{1} - __imagx);
+    const _Tp __term1_lo = ::cuda::std::fma(_Tp{1} - __imagx, _Tp{1} - __imagx, -__term1_hi) + __realx * __realx;
+
+    // Need (1.0 + __imagx)^2 with nearly full accuracy.
+    const _Tp __term2_sum_hi = (_Tp{1} + __imagx);
+    const _Tp __term2_sum_lo = ((_Tp{1} - __term2_sum_hi) + __imagx);
+
+    const _Tp __term2_sq_hi = __term2_sum_hi * __term2_sum_hi;
+    _Tp __term2_sq_lo       = ::cuda::std::fma(__term2_sum_hi, __term2_sum_hi, -__term2_sq_hi);
+    __term2_sq_lo += _Tp{2} * __term2_sum_hi * __term2_sum_lo;
+
+    // Multiple __term1_hi/lo and __term2_sq_hi/lo:
+    __inner_most_term_hi = __term1_hi * __term2_sq_hi;
+    __inner_most_term_lo = ::cuda::std::fma(__term1_hi, __term2_sq_hi, -__inner_most_term_hi);
+    // All needed:
+    __inner_most_term_lo += __term1_hi * __term2_sq_lo + __term1_lo * __term2_sq_hi;
+  }
+
+  // Normalize the above (assumed in the extended-sqrt function):
+  const _Tp __norm_hi  = __inner_most_term_hi + __inner_most_term_lo;
+  const _Tp __norm_lo  = -((__norm_hi - __inner_most_term_hi) - __inner_most_term_lo);
+  __inner_most_term_hi = __norm_hi;
+  __inner_most_term_lo = __norm_lo;
+
+  // Extended sqrt function:
+  // (__extended_sqrt_hi + __extended_sqrt_lo) = sqrt(__inner_most_term_hi + __inner_most_term_lo)
+  const __cccl_asinh_sqrt_return_hilo<_Tp> __extended_sqrt_hilo =
+    __internal_double_Tp_sqrt_unsafe<_Tp>(__inner_most_term_hi, __inner_most_term_lo);
+
+  _Tp __extended_sqrt_hi = __extended_sqrt_hilo.__hi;
+  _Tp __extended_sqrt_lo = __extended_sqrt_hilo.__lo;
+
+  // 0.0, and some very particular values, do not survive this unsafe sqrt function.
+  // This case occurs when (1 + x^2) is zero or denormal. (and rsqrt(x)*rsqrt(x) become inf).
+  constexpr __uint_t __min_normal_bits = __uint_t(0x1) << __mant_nbits;
+  const _Tp __min_normal               = ::cuda::std::bit_cast<_Tp>(__min_normal_bits);
+
+  if (__inner_most_term_hi <= _Tp{2} * __min_normal)
+  {
+    __extended_sqrt_hi = _Tp{2} * __realx;
+    __extended_sqrt_lo = _Tp{0};
+  }
+
+  // Get sqrt(0.5*(__extended_sqrt_hi + __diffx_m1))
+  // This can result in catastrophic cancellation if __diffx_m1 < 0, in this case
+  // We instead use the equivalent
+  //     (__realx*__imagx) / sqrt(0.5*(__extended_sqrt_hi - __diffx_m1))
+
+  const _Tp __inside_sqrt_term = _Tp{0.5} * (::cuda::std::fabs(__diffx_m1) + __extended_sqrt_hi);
+
+  // Allow for rsqrt optimization:
+  // We can have two slightly different paths depending on whether rsqrt is available
+  // or not, aka are we on device or host.
+
+  const _Tp __recip_sqrt = __internal_rsqrt_inverse_hyperbloic<_Tp>(__inside_sqrt_term);
+  _Tp __pos_evaluation_real;
+
+  // This reuses the sqrt calculated on CPU already in __recip_sqrt,
+  // And gets sqrt quickly on device using the rsqrt already calculated.
+#if _CCCL_CUDA_COMPILATION()
+  NV_IF_ELSE_TARGET(NV_IS_DEVICE,
+                    (__pos_evaluation_real = (__recip_sqrt * __inside_sqrt_term);),
+                    (__pos_evaluation_real = ::cuda::std::sqrt(__inside_sqrt_term);))
+#else
+  __pos_evaluation_real = ::cuda::std::sqrt(__inside_sqrt_term);
+#endif // _CCCL_CUDA_COMPILATION()
+
+  // Here, in a happy coincidence(?), we happen to intermediately calculate an accurate
+  // return value for the real part of the answer in the case that __realx is small,
+  // as you would obtain from the Taylor expansion of asinh. (~ real/sqrt(1 - imag^2)).
+  // The following parts of the calculation result in bad catastrophic cancellation for
+  // this case, so we save this intermediate value:
+  const _Tp __small_x_real_return_val = __realx * __recip_sqrt;
+  const _Tp __pos_evaluation_imag     = __imagx * __small_x_real_return_val;
+
+  const _Tp __sqrt_real_part = (__diffx_m1 > _Tp{0}) ? __pos_evaluation_real : __pos_evaluation_imag;
+  const _Tp __sqrt_imag_part = (__diffx_m1 > _Tp{0}) ? __pos_evaluation_imag : __pos_evaluation_real;
+
+  // For an accurate log, we calculate |(__sqrt_real_part + i*__sqrt_imag_part)| - 1 and use log1p.
+  // This can normally have bad catastrophic cancellation, however
+  // we have a lot of retained enough accuracy to subtract fairly simply:
+  const _Tp __m1  = __extended_sqrt_hi - _Tp{1};
+  const _Tp __rem = -((__m1 + _Tp{1}) - __extended_sqrt_hi);
+
+  __extended_sqrt_hi = __m1;
+  __extended_sqrt_lo += __rem;
+
+  // Final sum before sending it to log1p, all terms needed.
+  // Add our sum via three terms, adding equally sized components.
+  const _Tp __sum1 = (__x_abs_sq_hi + __extended_sqrt_hi);
+  const _Tp __sum2 = _Tp{2} * (__realx * __sqrt_real_part + __imagx * __sqrt_imag_part);
+  const _Tp __sum3 = (__extended_sqrt_lo + __x_abs_sq_lo);
+
+  const _Tp __abs_sqrt_part_sq = __sum1 + (__sum2 + __sum3);
+
+  const _Tp __atan2_input1 = __imagx + __sqrt_imag_part;
+  const _Tp __atan2_input2 = __realx + __sqrt_real_part;
+
+  _Tp __ans_real       = _Tp{0.5} * ::cuda::std::log1p(__abs_sqrt_part_sq);
+  const _Tp __ans_imag = ::cuda::std::atan2(__atan2_input1, __atan2_input2);
+
+  // The small |real| case, as mentioned above.
+  // Bounds found by testing.
+  _Tp __realx_small_bound_override = _Tp{2.220446e-16};
+
+  if constexpr (is_same_v<_Tp, float>)
+  {
+    __realx_small_bound_override = _Tp{6.0e-08f};
+  }
+
+  if (__realx < __realx_small_bound_override && __imagx < _Tp(1))
+  {
+    __ans_real = __small_x_real_return_val;
+  }
+
+  __ans_real += __x_big_factor;
+
+  // Copy signs back in
+  return complex<_Tp>(::cuda::std::copysign(__ans_real, __x.real()), ::cuda::std::copysign(__ans_imag, __x.imag()));
 }
 
 // We have performance issues with some trigonometric functions with extended floating point types
 #if _LIBCUDACXX_HAS_NVBF16()
 template <>
 _CCCL_API inline complex<__nv_bfloat16> asinh(const complex<__nv_bfloat16>& __x)
 {
-  return complex<__nv_bfloat16>{::cuda::std::asinh(complex<float>{__x})};
+  const complex<__nv_bfloat16> ans = complex<__nv_bfloat16>{::cuda::std::asinh(complex<float>{__x})};
+
+  return ans;
 }
 #endif // _LIBCUDACXX_HAS_NVBF16()